Gini Index and Polynomial Pen s Parade
|
|
- Kory James
- 6 years ago
- Views:
Transcription
1 Gii Idex ad Polyomial Pe s Parade Jules Sadefo Kamdem To cite this versio: Jules Sadefo Kamdem. Gii Idex ad Polyomial Pe s Parade <hal > HAL Id: hal Submitted o 2 Apr 2011 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.
2 Gii Idex ad Polyomial Pe's Parade Jules SADEFO KAMDEM Uiversité Motpellier I Lameta CNRS UMR 5474, FRANCE Abstract I this paper, we propose a simple way to compute the Gii idex whe icome y is a ite order k N polyomial fuctio of its rak amog idividuals. Key-words ad phrases: Gii, Icome iequality, Polyomial pe's parade, Raks. JEL Classicatio: D63, D31, C15. Correspodig author: Uiversité de Motpellier I, UFR Scieces Ecoomiques, Aveue Raymod Dugrad - Site Richter - CS 79606, F Motpellier Cedex 2, Frace; sadefo@lameta.uiv-motp1.fr 1
3 1 Itroductio Iterest i the lik betwee icome ad its rak is kow i the icome distributio literature as Pe's parade followig Pe (1971, 1973). The precede has motivated research o the relatioship betwee Pe's parade ad the Gii idex that is a very importat iequality measure. However, this research has so far focused o the liear Pe's parade for which icome icreases by a costat amout as its rak icreases by oe uit (see Milaovic (1997). However, a liear parade does ot closely t may real world icome distributios, Pe's parade, which is covex i the absece of egative icomes (i.e., icomes that icrease by a greater amout as its rak icreases by each additioal uit). Mussard et al. (2010) has recetly itroduced the computatio of Gii idex with a covex quadratic Pe's parade (or secod degree polyomial) parade for which icome is a quadratic fuctio of its rak. I this paper, we exted the computatio of gii idex by usig a more geeral ad empirically more realistic case i which Pe's parade is a covex polyomial of ite order k N. Hece, the Gii idexes for a liear Pe's parade ad for quadratic pe's parade becomes a special case of that for a higher degree polyomial Pe's parade uder some costraits o parameters to keep covexity. The rest of the paper is orgaized as follows: I Sectio 2, the specicatio for a higher degree polyomial Pe's parade is provided. I Sectio 3, the problem of ttig a higher degree polyomial Pe's parade to real world data sets of Milaovic (1997) is discussed. Cocludig is i Sectio 4. 2 High degree Polyomial Pe's Parade Suppose that positive icomes, expressed as a vector y, deped o idividuals' raks r y i ay give icome distributio of size. Suppose that icomes are raked i ascedig order ad let r y 1 for the poorest idividual ad r y for the richest oe. Hece, followig Lerma ad Yitzhaki (1984), the Gii idex may be rewritte as follows: G 2 cov(y, r y) ȳ. (1) 2
4 Here, cov(y, r y ) represets the covariace betwee icomes ad raks ad ȳ the mea icome. It is straightforward to rewrite (13) as: G 2 σ y σ ry ρ(y, r y ) ȳ, (2) where ρ(y, r y ) is Pearso's correlatio coeciet betwee icomes y ad idividuals' raks r y, where σ y is the stadard deviatio of y ad where σ ry is the stadard deviatio of r y. Followig (2) ad uder the assumptio of a liear Pe's parade (i.e. y a + b r y ), Milaovic (1997) demostrates that for a sucietly large, the Gii idex ca be further expressed as: G σ y ρ(y, r y ). (3) 3y Milaovic's result is very iterestig sice it yields a simple way to compute the Gii idex. However, as metioed by Milaovic (1997, page 48) himself, "i almost all real world cases, Pe's parade is covex: icomes rise very slowly at the begiig, the go up by their absolute icrease, ad ally icrease eve at the rate of acceleratio". Thus, ρ(y, r y ) which measures liear correlatio will be less tha 1. Agai, from Milaovic (1997), a covex Pe's parade may be derived from a liear Pe's parade throughout regressive trasfers (poor-to-rich icome trasfers). Ispired by Milaovic's dig, we demostrate i the sequel, without takig recourse to regressive trasfers, that the Gii idex ca be computed with a geeral oliear polyomial fuctio Pe's parade. The computatio of the Gii Idex usig a polyomial fuctio of order 2 (i.e. quadratic fuctio) Pe's Parade has bee itroduced i Mussard et al.(2010). I this paper, we geeralize the precedure to compute the Gii Idex usig a polyomial fuctio of order k N pe's parade. Note that a liear pe's parade correspods to k 1 ad quadratic pe's parade correspods to k 2. Our result here will be available for each ite k N. 2.1 Simple Gii Idex with oliear power Pe's Parade Cosider a power fuctio relatio betwee icomes ad raks: k 1 y b i ry i + b k ry k. (4) 3
5 with k N. The covariace betwee y ad r j y for j N is give by: cov(y, r j y) b i cov ( ) ry, i ry j b i b i ri+j+1 y r y1 b i 2 r i+j+1 y ry j ry i r y1 r y1 b i ry i ry j r y1 r y1 (5) The mea icome ȳ is the: [ y b i ry i 1 ] b i r i (6) 2.2 The coeciet of variatio Sice the icomes y are positive, we use (13) by assumig that b k > 0 ad b j for j 1,..., k 1 are chose such that y > 0. For istace, if k 2 the we ca use b j for j 1,..., k such that b 2 1 4b 2 b 0 < 0. We are ow able to compute the coeciet of variatio of icomes as follows: σ y y k j0 b ib j cov(r i y, r j y) b i r i y (7) k b i j0 b j cov(r i y, r j y) b i r i y (8) k j0 b j cov(y, r j y) b i r i y (9) k j0 b j [ b i r y1 ri+j+1 y 1 [ b i ] 2 r y1 ri y r y1 rj y ( k b ir i )]. 4
6 where the variace of r k y is cov(r k y, r k y) σ 2 r k y 1 r 2k ( 1 the covariace betwee r j y ad r i y for 0 i, j k, is: cov ( ) ry, i ry j 1 After a double summatio ( ) cov b i ry, i b j ry j j0 r i+j 1 2 b i b j [ 1 ) 2 r k, (10) r i r j. (11) r i+j 1 2 r i ] r j. (12) Lemma 2.1 Whe, for q N ad r N, we have that r q q+1 q + 1. (13) Based o the precedig lemma, the variace of y whe is equivalet to: ( ) cov ry, i ry j b2 k k+k+1 k + k + 1 b2 k k+1 k+1 2 k + 1 k + 1 ( k b k k) 2 (2k + 1)(k + 1). 2 j0 (14) Therefore whe the stadard deviatio of y is equivalet to ( σ y k b k k) 2 (2k + 1)(k + 1) k b k 2 (k + 1) 2k + 1 k. (15) Whe, the mea of y is equivalet to y 1 b i r i b k k+1 k k + 1 b k k + 1 Thereby, as the coeciet of variatio is is equivalet expressed as: (16) σ y y k b k (k+1) 2k+1 k b k k k+1 (17) 5
7 the we have the followig limit which depeds o k ad the sig of b k : σ y lim y b k We have the proved the followig theorem: b k k k sig(b k ). (18) 2k + 1 2k + 1 Theorem 2.1 Uder the assumptio of a oliear polyomial Pe's parade, i.e., y b i ry, i with b k 0, whe, the coeciet of variatio of the reveue y has the followig limit: σ y lim y b k b k k (19) 2k + 1 O the other had, followig Milaovic (1997): lim 2 σ r y lim (20) The product of (20), (18) ad ρ(y, r y ) etails the followig result: Theorem 2.2 Uder the assumptio of a oliear polyomial Pe's parade, i.e., y b i ry, i with b k 0, whe, the Gii idex G k ca be approximately compute as follows: G k 1 b k k ρ(y, r y ), (21) 3 2k + 1 b k where ρ(y, r y ) is the correlatio coeciet betwee the reveue ad the rak r y. Remark 2.1 Note that for k 1, we have the result of Milaovic (1997), i.e. G 1 ρ(y,ry) ad for k 2, we have the result of Sadefo Kamdem et al. 3 (2010), i.e. G 2 2 ρ(y,ry) 15. Remark 2.2 Remarks that the Gii Idex G k approximatio depeds o k, the sig of b k ad ρ(y, r y ). Sice the Gii computatio is idepedet of parameters b i for i 0,..., k 1, we ca compute the Pe's Parade as follows: y b 0 + b k r k. Based o reveue data, it is simple to use a regressio to estimate the parameters b 0 ad b k. 6
8 3 Applicatio with Milaovic (1997) Data I relatio with the paramater k ad b k > 0, we have the followig table: Table 1: Computatio of some coeciet of variatio GV k. k CV k 0,577 0,894 1,134 1,333 1,508 1,664 1,807 1,940 GC k where CV k deotes the coeciet of variatio for polyomial Pe's Parade of order k ad GC k CV k / 3. Followig Milaovic's data (1997), we obtai the followig results: Table 2: Compariso the gii idexes of G k for k 1, 2, 3, 6 with true G est Coutry (year) ρ(y, r y ) G 1 G 2 G 3 G 6 G est Hugary (1993; aual) Polad (1993; aual) Romaia (1994; mothly) Bulgaria (1994; aual) Estoia(1995; quarterly) UK (1986; aual) Germay (1889; aual) US (1991; aual) Russia (1993-4; quarterly)) Kyrgyzsta (1993; quarterly) I the precedig table, G est deotes the estimatio of the true Gii idex by usig Milaovic data (1997). Remark 3.1 Based o the aalysis of the precedig table, we propose to cosider liear Pe's parade (k1) to compute the Gii idex of Polad (1993, aual), Romaia (1994, mothly), Bulgaria(1994; aual). For estoia (1995; quarterly) ad UK (1986; aual), we ca also choose k 1, but we cosider k 2 i the case where govermet policy prefers to overestimate iequality istead of uderestimate iequality. For Russia (1993-4; quarterly), we cosider k 3 ad for Kyrgyzsta (1993; quarterly) we choose k 6. 7
9 Remark 3.2 I our polyomial pe's parade, if r y 1, the y y 1 i1 b k. I practical applicatios, the parameters b k, estimated from the observed data usig multiple regressio, are chose so that the reveue y > 0 (i.e. b k > 0). It's very importat to ote that, the correlatio coeciet betwee y ad its rak r y deped o b k. Igorig this fact will arbitrarily restrict the Parade to pass through the origi ad may result i less accurate estimates of the Gii Idex. 4 Cocludig Remarks Followig Milaovic (1997), we have proposed aother simple way to calculate the Gii coeciet uder the assumptio of a geeral polyomial Pe's Parade of order k. By usig the data i Table 2, we coclude that the computatio of the Gii Idex of each reveues data eeds to d a specic iteger k, which is the order of a specic polyomial Pe's Parade. Our Gii computatio is useful i practical applicatios as soo as the limit expressio obtaied i (21)is a good approximatio of the Gii Idex for usual size. Two immediate ad practical applicatios ca be geerated from this ew Gii expressio. First, the possibility to address a sigicat test sice our Gii Idex (as well as Milaovic's) is based o Pearso's correlatio coeciet. Thereby, testig for the Gii Idex sigicace is equivalet to testig for the sigicace of Pearso's correlatio coeciet (up to the k costat 2k+1 ). This test relies o the well-kow studet statistics based o the hyperbolic taget trasformatio. Secod, estimatig the coeciets b i for i 1,..., k, e.g. with Yitzhaki's Gii regressio aalysis or a multiple regressio, eables a parametric Gii Idex to be obtaied. That procedure depeds o parameters reectig the curvature of Pe's Parade. This may be of iterest whe oe compares the shape of two icome distributios. 8
10 Refereces [1] Milaovic, B., "A simple way to calculate the Gii coeciet, ad some implicatios," Ecoomics Letters, 56, 45-49, [2] Lerma R.I., S. Yitzhaki, "A ote o the calculatio ad iterpretatio of the Gii idex," Ecoomics Letters, 15, , [3] Pe, J., Icome Distributio. (Alle Lae The Pegui Press, Lodo), [4] Pe, J., A parade of dwarfs (ad a few giats),i: A.B. Atkiso, ed., Wealth, Icome ad Iequality: Selected Readigs, (Pegui Books, Middlesex) 73-82, [5] Mussard, S., Sadefo Kamdem, J., Seyte, F., Terraza, M., "Quadratic Pe's parade ad the computatio of Gii Idex, Accepted for publicatio i Review of Icome ad Wealth,
Coefficient of variation and Power Pen s parade computation
Coefficiet of variatio ad Power Pe s parade computatio Jules Sadefo Kamdem To cite this versio: Jules Sadefo Kamdem. Coefficiet of variatio ad Power Pe s parade computatio. 20. HAL Id: hal-0058658
More informationGroupe de Recherche en Économie et Développement International. Cahier de Recherche / Working Paper 10-18
Groupe de Recherche e Écoomie et Développemet Iteratioal Cahier de Recherche / Workig Paper 0-8 Quadratic Pe's Parade ad the Computatio of the Gii idex Stéphae Mussard, Jules Sadefo Kamdem Fraçoise Seyte
More informationImprovement of Generic Attacks on the Rank Syndrome Decoding Problem
Improvemet of Geeric Attacks o the Rak Sydrome Decodig Problem Nicolas Arago, Philippe Gaborit, Adrie Hauteville, Jea-Pierre Tillich To cite this versio: Nicolas Arago, Philippe Gaborit, Adrie Hauteville,
More informationOn the behavior at infinity of an integrable function
O the behavior at ifiity of a itegrable fuctio Emmauel Lesige To cite this versio: Emmauel Lesige. O the behavior at ifiity of a itegrable fuctio. The America Mathematical Mothly, 200, 7 (2), pp.75-8.
More informationThe Goldbach conjectures
The Goldbach cojectures Jamel Ghaouchi To cite this versio: Jamel Ghaouchi. The Goldbach cojectures. 2015. HAL Id: hal-01243303 https://hal.archives-ouvertes.fr/hal-01243303 Submitted o
More informationA Simple Proof of the Shallow Packing Lemma
A Simple Proof of the Shallow Packig Lemma Nabil Mustafa To cite this versio: Nabil Mustafa. A Simple Proof of the Shallow Packig Lemma. Discrete ad Computatioal Geometry, Spriger Verlag, 06, 55 (3), pp.739-743.
More informationOptimization Results for a Generalized Coupon Collector Problem
Optimizatio Results for a Geeralized Coupo Collector Problem Emmauelle Aceaume, Ya Busel, E Schulte-Geers, B Sericola To cite this versio: Emmauelle Aceaume, Ya Busel, E Schulte-Geers, B Sericola. Optimizatio
More informationTURBULENT FUNCTIONS AND SOLVING THE NAVIER-STOKES EQUATION BY FOURIER SERIES
TURBULENT FUNCTIONS AND SOLVING THE NAVIER-STOKES EQUATION BY FOURIER SERIES M Sghiar To cite this versio: M Sghiar. TURBULENT FUNCTIONS AND SOLVING THE NAVIER-STOKES EQUATION BY FOURIER SERIES. Iteratioal
More informationInvariant relations between binary Goldbach s decompositions numbers coded in a 4 letters language
Ivariat relatios betwee biary Goldbach s decompositios umbers coded i a letters laguage Deise Vella-Chemla To cite this versio: Deise Vella-Chemla. Ivariat relatios betwee biary Goldbach s decompositios
More informationTesting the number of parameters with multidimensional MLP
Testig the umber of parameters with multidimesioal MLP Joseph Rykiewicz To cite this versio: Joseph Rykiewicz. Testig the umber of parameters with multidimesioal MLP. ASMDA 2005, 2005, Brest, Frace. pp.561-568,
More informationA RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS
J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider k-sample ad chage poit problems for idepedet data i a
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationGUIDE FOR THE USE OF THE DECISION SUPPORT SYSTEM (DSS)*
GUIDE FOR THE USE OF THE DECISION SUPPORT SYSTEM (DSS)* *Note: I Frech SAD (Système d Aide à la Décisio) 1. Itroductio to the DSS Eightee statistical distributios are available i HYFRAN-PLUS software to
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.
More informationDecomposition of Gini and the generalized entropy inequality measures. Abstract
Decompositio of Gii ad the geeralized etropy iequality measures Mussard Stéphae LAMETA Uiversity of Motpellier I Terraza Michel LAMETA Uiversity of Motpellier I Seyte Fraçoise LAMETA Uiversity of Motpellier
More informationExplicit Maximal and Minimal Curves over Finite Fields of Odd Characteristics
Explicit Maximal ad Miimal Curves over Fiite Fields of Odd Characteristics Ferruh Ozbudak, Zülfükar Saygı To cite this versio: Ferruh Ozbudak, Zülfükar Saygı. Explicit Maximal ad Miimal Curves over Fiite
More informationPrinciple Of Superposition
ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give
More informationEfficient GMM LECTURE 12 GMM II
DECEMBER 1 010 LECTURE 1 II Efficiet The estimator depeds o the choice of the weight matrix A. The efficiet estimator is the oe that has the smallest asymptotic variace amog all estimators defied by differet
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationNEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
More informationSummary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.
Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationTHE KALMAN FILTER RAUL ROJAS
THE KALMAN FILTER RAUL ROJAS Abstract. This paper provides a getle itroductio to the Kalma filter, a umerical method that ca be used for sesor fusio or for calculatio of trajectories. First, we cosider
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by
More informationA note on the sum of uniform random variables
A ote o the sum of uiform radom variables Aiello Buoocore, Erica Pirozzi, Luigia Caputo To cite this versio: Aiello Buoocore, Erica Pirozzi, Luigia Caputo. A ote o the sum of uiform radom variables. Statistics
More informationA 2nTH ORDER LINEAR DIFFERENCE EQUATION
A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy
More informationSTATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. Comments:
Recall: STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Commets:. So far we have estimates of the parameters! 0 ad!, but have o idea how good these estimates are. Assumptio: E(Y x)! 0 +! x (liear coditioal
More informationStatistical Properties of OLS estimators
1 Statistical Properties of OLS estimators Liear Model: Y i = β 0 + β 1 X i + u i OLS estimators: β 0 = Y β 1X β 1 = Best Liear Ubiased Estimator (BLUE) Liear Estimator: β 0 ad β 1 are liear fuctio of
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationEconomics 241B Relation to Method of Moments and Maximum Likelihood OLSE as a Maximum Likelihood Estimator
Ecoomics 24B Relatio to Method of Momets ad Maximum Likelihood OLSE as a Maximum Likelihood Estimator Uder Assumptio 5 we have speci ed the distributio of the error, so we ca estimate the model parameters
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationChapter 5. Inequalities. 5.1 The Markov and Chebyshev inequalities
Chapter 5 Iequalities 5.1 The Markov ad Chebyshev iequalities As you have probably see o today s frot page: every perso i the upper teth percetile ears at least 1 times more tha the average salary. I other
More informationSOME TRIBONACCI IDENTITIES
Mathematics Today Vol.7(Dec-011) 1-9 ISSN 0976-38 Abstract: SOME TRIBONACCI IDENTITIES Shah Devbhadra V. Sir P.T.Sarvajaik College of Sciece, Athwalies, Surat 395001. e-mail : drdvshah@yahoo.com The sequece
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationA New Solution Method for the Finite-Horizon Discrete-Time EOQ Problem
This is the Pre-Published Versio. A New Solutio Method for the Fiite-Horizo Discrete-Time EOQ Problem Chug-Lu Li Departmet of Logistics The Hog Kog Polytechic Uiversity Hug Hom, Kowloo, Hog Kog Phoe: +852-2766-7410
More informationGUIDELINES ON REPRESENTATIVE SAMPLING
DRUGS WORKING GROUP VALIDATION OF THE GUIDELINES ON REPRESENTATIVE SAMPLING DOCUMENT TYPE : REF. CODE: ISSUE NO: ISSUE DATE: VALIDATION REPORT DWG-SGL-001 002 08 DECEMBER 2012 Ref code: DWG-SGL-001 Issue
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should be doe
More informationLecture 22: Review for Exam 2. 1 Basic Model Assumptions (without Gaussian Noise)
Lecture 22: Review for Exam 2 Basic Model Assumptios (without Gaussia Noise) We model oe cotiuous respose variable Y, as a liear fuctio of p umerical predictors, plus oise: Y = β 0 + β X +... β p X p +
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationStatistical Inference Based on Extremum Estimators
T. Rotheberg Fall, 2007 Statistical Iferece Based o Extremum Estimators Itroductio Suppose 0, the true value of a p-dimesioal parameter, is kow to lie i some subset S R p : Ofte we choose to estimate 0
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationDimension-free PAC-Bayesian bounds for the estimation of the mean of a random vector
Dimesio-free PAC-Bayesia bouds for the estimatio of the mea of a radom vector Olivier Catoi CREST CNRS UMR 9194 Uiversité Paris Saclay olivier.catoi@esae.fr Ilaria Giulii Laboratoire de Probabilités et
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationStudy the bias (due to the nite dimensional approximation) and variance of the estimators
2 Series Methods 2. Geeral Approach A model has parameters (; ) where is ite-dimesioal ad is oparametric. (Sometimes, there is o :) We will focus o regressio. The fuctio is approximated by a series a ite
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationLecture 3. Properties of Summary Statistics: Sampling Distribution
Lecture 3 Properties of Summary Statistics: Samplig Distributio Mai Theme How ca we use math to justify that our umerical summaries from the sample are good summaries of the populatio? Lecture Summary
More informationECON 3150/4150, Spring term Lecture 3
Itroductio Fidig the best fit by regressio Residuals ad R-sq Regressio ad causality Summary ad ext step ECON 3150/4150, Sprig term 2014. Lecture 3 Ragar Nymoe Uiversity of Oslo 21 Jauary 2014 1 / 30 Itroductio
More informationSession 5. (1) Principal component analysis and Karhunen-Loève transformation
200 Autum semester Patter Iformatio Processig Topic 2 Image compressio by orthogoal trasformatio Sessio 5 () Pricipal compoet aalysis ad Karhue-Loève trasformatio Topic 2 of this course explais the image
More informationSingle Crossing Lorenz Curves and Inequality Comparisons
Sigle Crossig Lorez Curves ad Iequality Comparisos Thibault Gajdos To cite this versio: Thibault Gajdos. Sigle Crossig Lorez Curves ad Iequality Comparisos. Mathematical Social Scieces, Elsevier, 2004,
More informationResampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More information11 Correlation and Regression
11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record
More informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationx a x a Lecture 2 Series (See Chapter 1 in Boas)
Lecture Series (See Chapter i Boas) A basic ad very powerful (if pedestria, recall we are lazy AD smart) way to solve ay differetial (or itegral) equatio is via a series expasio of the correspodig solutio
More informationBull. Korean Math. Soc. 36 (1999), No. 3, pp. 451{457 THE STRONG CONSISTENCY OF NONLINEAR REGRESSION QUANTILES ESTIMATORS Seung Hoe Choi and Hae Kyung
Bull. Korea Math. Soc. 36 (999), No. 3, pp. 45{457 THE STRONG CONSISTENCY OF NONLINEAR REGRESSION QUANTILES ESTIMATORS Abstract. This paper provides suciet coditios which esure the strog cosistecy of regressio
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationPrecise Rates in Complete Moment Convergence for Negatively Associated Sequences
Commuicatios of the Korea Statistical Society 29, Vol. 16, No. 5, 841 849 Precise Rates i Complete Momet Covergece for Negatively Associated Sequeces Dae-Hee Ryu 1,a a Departmet of Computer Sciece, ChugWoo
More informationNon-parametric investigation of the Kuznets hypothesis for transitional countries
Abstract No-parametric ivestigatio of the Kuzets hypothesis for trasitioal coutries Demidova Olga State Uiversity Higher School of Ecoomics The Kuzets hypothesis states that the relatio betwee icome distributio
More informationInformation-based Feature Selection
Iformatio-based Feature Selectio Farza Faria, Abbas Kazeroui, Afshi Babveyh Email: {faria,abbask,afshib}@staford.edu 1 Itroductio Feature selectio is a topic of great iterest i applicatios dealig with
More informationCorrelation Regression
Correlatio Regressio While correlatio methods measure the stregth of a liear relatioship betwee two variables, we might wish to go a little further: How much does oe variable chage for a give chage i aother
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More informationA note on self-normalized Dickey-Fuller test for unit root in autoregressive time series with GARCH errors
Appl. Math. J. Chiese Uiv. 008, 3(): 97-0 A ote o self-ormalized Dickey-Fuller test for uit root i autoregressive time series with GARCH errors YANG Xiao-rog ZHANG Li-xi Abstract. I this article, the uit
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More informationGG313 GEOLOGICAL DATA ANALYSIS
GG313 GEOLOGICAL DATA ANALYSIS 1 Testig Hypothesis GG313 GEOLOGICAL DATA ANALYSIS LECTURE NOTES PAUL WESSEL SECTION TESTING OF HYPOTHESES Much of statistics is cocered with testig hypothesis agaist data
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
More informationNumber of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationGoodness-Of-Fit For The Generalized Exponential Distribution. Abstract
Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
More informationREGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data
More informationFast Exact Filtering in Generalized Conditionally Observed Markov Switching Models with Copulas
Fast Exact Filterig i Geeralized Coditioally Observed Markov Switchig Models with Copulas Fei Zheg, Stéphae Derrode, Wojciech Pieczyski To cite this versio: Fei Zheg, Stéphae Derrode, Wojciech Pieczyski.
More informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationBayesian Methods: Introduction to Multi-parameter Models
Bayesia Methods: Itroductio to Multi-parameter Models Parameter: θ = ( θ, θ) Give Likelihood p(y θ) ad prior p(θ ), the posterior p proportioal to p(y θ) x p(θ ) Margial posterior ( θ, θ y) is Iterested
More information9. Simple linear regression G2.1) Show that the vector of residuals e = Y Ŷ has the covariance matrix (I X(X T X) 1 X T )σ 2.
LINKÖPINGS UNIVERSITET Matematiska Istitutioe Matematisk Statistik HT1-2015 TAMS24 9. Simple liear regressio G2.1) Show that the vector of residuals e = Y Ŷ has the covariace matrix (I X(X T X) 1 X T )σ
More informationRegression with an Evaporating Logarithmic Trend
Regressio with a Evaporatig Logarithmic Tred Peter C. B. Phillips Cowles Foudatio, Yale Uiversity, Uiversity of Aucklad & Uiversity of York ad Yixiao Su Departmet of Ecoomics Yale Uiversity October 5,
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationSection 13.3 Area and the Definite Integral
Sectio 3.3 Area ad the Defiite Itegral We ca easily fid areas of certai geometric figures usig well-kow formulas: However, it is t easy to fid the area of a regio with curved sides: METHOD: To evaluate
More informationDouble Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution
Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.
More informationSummary and Discussion on Simultaneous Analysis of Lasso and Dantzig Selector
Summary ad Discussio o Simultaeous Aalysis of Lasso ad Datzig Selector STAT732, Sprig 28 Duzhe Wag May 4, 28 Abstract This is a discussio o the work i Bickel, Ritov ad Tsybakov (29). We begi with a short
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationMachine Learning Brett Bernstein
Machie Learig Brett Berstei Week 2 Lecture: Cocept Check Exercises Starred problems are optioal. Excess Risk Decompositio 1. Let X = Y = {1, 2,..., 10}, A = {1,..., 10, 11} ad suppose the data distributio
More informationRelations between the continuous and the discrete Lotka power function
Relatios betwee the cotiuous ad the discrete Lotka power fuctio by L. Egghe Limburgs Uiversitair Cetrum (LUC), Uiversitaire Campus, B-3590 Diepebeek, Belgium ad Uiversiteit Atwerpe (UA), Campus Drie Eike,
More information4.3 Growth Rates of Solutions to Recurrences
4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.
More informationDiscrete-Time Systems, LTI Systems, and Discrete-Time Convolution
EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [
More informationThe Principle of Strong Diminishing Transfer
The Priciple of Strog Dimiishig Trasfer Alai Chateaueuf, Thibault Gajdos, Pierre-Hery Wilthie To cite this versio: Alai Chateaueuf, Thibault Gajdos, Pierre-Hery Wilthie. The Priciple of Strog Dimiishig
More informationCEU Department of Economics Econometrics 1, Problem Set 1 - Solutions
CEU Departmet of Ecoomics Ecoometrics, Problem Set - Solutios Part A. Exogeeity - edogeeity The liear coditioal expectatio (CE) model has the followig form: We would like to estimate the effect of some
More informationo <Xln <X2n <... <X n < o (1.1)
Metrika, Volume 28, 1981, page 257-262. 9 Viea. Estimatio Problems for Rectagular Distributios (Or the Taxi Problem Revisited) By J.S. Rao, Sata Barbara I ) Abstract: The problem of estimatig the ukow
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationFLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS. H. W. Gould Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 FLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS H. W. Gould Departmet of Mathematics, West Virgiia Uiversity, Morgatow, WV
More informationt distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference
EXST30 Backgroud material Page From the textbook The Statistical Sleuth Mea [0]: I your text the word mea deotes a populatio mea (µ) while the work average deotes a sample average ( ). Variace [0]: The
More informationProblem Set 4 Due Oct, 12
EE226: Radom Processes i Systems Lecturer: Jea C. Walrad Problem Set 4 Due Oct, 12 Fall 06 GSI: Assae Gueye This problem set essetially reviews detectio theory ad hypothesis testig ad some basic otios
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationCSE 527, Additional notes on MLE & EM
CSE 57 Lecture Notes: MLE & EM CSE 57, Additioal otes o MLE & EM Based o earlier otes by C. Grat & M. Narasimha Itroductio Last lecture we bega a examiatio of model based clusterig. This lecture will be
More information